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Introduction to the Laplace Transform: Definitions, Properties, and Applications in DEs

This chapter covers the fundamentals of the Laplace Transform, including its definition, basic properties, and methods for solving differential equations (DEs) using both the Laplace and inverse Laplace transforms. You'll learn how to compute the Laplace Transform for various functions, apply theorems related to derivatives, and utilize operational properties effectively. The content is structured to facilitate understanding and application in solving initial value problems through practical examples and exercises.

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Introduction to the Laplace Transform: Definitions, Properties, and Applications in DEs

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  1. THE LAPLACE TRANSFORM Chapter 4

  2. Plan I - Definition and basic properties II - Inverse Laplace transform and solutions of DE III - Operational Properties

  3. I – Definitions and basic properties Learning objective At the end of the lesson you should be able to : • Define Laplace Transform. • Find the Laplace Transform of different type of functions using the definition.

  4. Definition: Laplace Transform Let f be a function defined for Then the integral is said to be the Laplace transform of f, provided that the integral converges.

  5. Notations

  6. Use the definition to find the values of the following: Example 1

  7. Solution

  8. Solution

  9. Theorem: Transforms of some Basic Functions

  10. is a Linear Transform

  11. Example2 Find the Laplace transform of the function

  12. Example2

  13. Transform of a Piecewise function Example 3 Given Find

  14. Solution

  15. Laplace Transform of a Derivative Let Find

  16. Laplace Transform of a Derivative

  17. Laplace Transform of a Derivative Theorem where

  18. Laplace Transform of a Derivative Example Find the Laplace transform of the following IVP

  19. Laplace Transform of a Derivative Solution

  20. Laplace Transform of a Derivative Solution

  21. II – Inverse Laplace Transform and solutions of DEs Learning objective At the end of the lesson you should be able to : • Define Inverse Laplace Transform. • Solve ODEs using the Laplace Transform.

  22. Inverse Transforms If F (s) represents the Laplace transform of a function f (t), i.e., L {f (t)}=F (s) then f (t) is the inverse Laplace transform of F (s) and,

  23. Theorem : Some Inverse Transforms

  24. Application

  25. is a Linear Transform Where F and G are the transforms of some functions f and g .

  26. Division and Linearity Find

  27. Division and Linearity

  28. Partial Fractions in Inverse Laplace Find

  29. Partial Fractions in Inverse Laplace ,

  30. Example 1 Solve the partial given IVP by Laplace transform.

  31. Solution 1

  32. Solution 1

  33. Solution 1

  34. III – Operational Properties Learning objective At the end of the lesson you should be able to use translation theorems.

  35. First translation theorem If and is any real number, then .

  36. First translation theorem Example 1: .

  37. First translation theorem Example 2: .

  38. Inverse form of First translation theorem .

  39. Inverse form of First translation theorem Example 1: .

  40. Exercise Solve

  41. Solution

  42. Solution

  43. Solution

  44. Solution

  45. Unit Step Function or Heaviside Function The unit step function is defined as U 1 t

  46. Example What happen when is multiplied by the Heaviside function

  47. Example f f t 0 0 2 t -3 -3

  48. The Second Translation Theorem If and then

  49. Example 1 Let then

  50. Example 2 Find where

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